In many applications, one wishes to measure the phase of a sinusoidal signal relative to some point in time or another signal. Consider a sinusoid of the formV1(t)=A1 cos(ω1t+θ1)  (1)
The phase that is measured when the signal is measured is the instantaneous phase given byΦ1(t)=ω1t+γ1  (2)
This phase depends on the time at which the signal is measured. In general, the difference in phase between two signals is the quantity of interest. Denote the second signal byV2(t)=A2 cos(ω2t+θ2)  (3)
The phase difference Φ2−Φ1=(ω2−ω1)t+(θ2−θ1). This quantity is only time invariant if the frequencies of the two signals are the same. Hence, defining a phase difference that is independent of the time at which the signals are measured presents challenges if the frequencies of the two signals are different from one another.